Weak Lipschitz structures and their connections with the topological structures

TL;DR

This paper introduces weak Lipschitz structures using weak pseudo-metrics, explores their connection to topological structures, and studies the properties of weak Lipschitz uniformities and maps.

Contribution

It proposes a new concept of weak Lipschitz structures based on weak pseudo-metrics and establishes their relationship with topological and uniform structures.

Findings

Weak Lipschitz structures can define all topological structures.

Weak Lipschitz maps are uniformly continuous with respect to weak Lipschitz uniformities.

Connections between continuous maps and weak Lipschitz maps are established.

Abstract

Two approaches to Lipschitz structures for any set are presented, studied and compared. The first approach is similar to the one proposed in Fraser, Jr. R. B., Axiom systems for Lipschitz structures, Fundamenta Mathematicae, (1970), where Lipschitz structures are defined as families of pseudo-metrics satisfying suitable conditions. The other one, here introduced, is expressed by using weak pseudo-metrics, which (unlike the pseudo-metrics) do not necessarily vanish on the whole of the diagonal of the cartesian product of the considered set. In this case we will talk about weak Lipschitz structures. Since all topological structures are defined by a a family of weak pseudo-metrics (as we will show in Section 4) we can find some connections between topological structures and weak Lipschitz structures, and a link between continuous maps and weak Lipschitz maps. A central part of this paper is devoted to the weak Lipschitz uniformity defined by a weak Lipschitz structure, which is introduced in Section 8. A notion of uniform continuity with respect to weak Lipschitz uniformities is proposed and studied. In particular, we prove that the weak Lipschitz maps acting between two weak Lipschitz spaces are uniformly continuous with respect to the weak Lipschitz uniformities defined by the respective weak Lipschitz structures.